The article has absolutely no in-line cites. There are about four references listed in the References sections, but they might be furhter reading for all the averager reader can know. The prose is extraordinatily technical and certainly not accessible to the lay reader, as required by FA criteria. --Francesco Franco aka Lacatosias 10:00, 17 August 2006 (UTC)

No inline-cites: Common style in academic writing (at least in maths and related subjects) is not to provide references for well-known facts. Most, if not all, of the article is well known and can be found in any linear algebra book. I haven't followed Wikipedia policies on this subject, but if it has become accepted that every fact should have an inline reference, then I'd like to know.

References section: I do not understand this comment. Should it be mentioned after every reference, that this has in fact been used? I thought that materials for further reading should go in a "Further reading" section.

Too technical: Which FA criterion says that it should be accessible to the lay reader? I agree that some parts will be hard to understand for somebody without a scientific background, but that's inevitable in my opinion. I don't see where the prose is too technical; could you give an example? -- Jitse Niesen (talk) 10:33, 17 August 2006 (UTC)

I second what Jitse said about inline citations: they're quite useless in the context, as it is common math knowledge that can be found in any ordinary algebra book. And yes, this is a math article, yet it explains a quite complicated concept in a way that appears very simple to me. In short I don't see any problems... -- Grafikm(AutoGRAF) 11:39, 17 August 2006 (UTC)

Please check WP:WIAFA - no criterion says that it should be accesible to the lay reader. Besides, this is one of the most comprehensive presentations of math I've ever seen. Even as a lay man, I'm sure that you can get a very good idea what eigenvectors are and this is more than one should hope from a 5 min. read. AdamSmithee 11:10, 17 August 2006 (UTC)

Oh, sure, sure. You folks don't have to do jack-shit basically. That's what you're telling me, eh?? I have to find 6000 references for a philosophy article which cinatined nothign but accted facts about what the person put forth, wheen he did, whi disgreed with him and so on. These thing are all accepted facts withing the philosopcial community.

First of all, cool off. Basically, folks do not have to do "jack-shit" unless they want to because this is unpaid voluntary work AdamSmithee 13:50, 17 August 2006 (UTC)

Heres the relevant section on prose with which I was lambasted in my FAC:

The perfect article “is understandable; it is clearly expressed for both experts and non-experts in appropriate detail, and thoroughly explores and explains the subject.”

Wiki is not a textbook. Prose should be accessible to laypersons and non-experts."

Seriously, some parts of this article sound exctly like my "linear algebra" textbook. There are no cites. This is an actionable objection, I think. --Francesco Franco aka Lacatosias 12:03, 17 August 2006 (UTC)

Then, could you be more specific on what parts do you find hard to follow for the layman? However, take into account that people should not get scared of the word vector when they read about eigenvectors. If it would be possible for people to go from no mathematics knowledge to full understanding of eigenvectors after reading an article for 5 mins. everyone would be a math Ph.D. before age 10. AdamSmithee 13:50, 17 August 2006 (UTC)

I have to agree that the article should have more inline citations - especially in the "Applications" section, but this is not the main issue with this article. Currently the article reads like a math textbook and probably belongs more to Wikibooks or Wikiversity than Wikipedia. What needs to be written in order for it to become encyclopaedic is a section on the development of eigenvalues. This means something about the creation of eigenvalues and how it has evolved to become what we know today. There should also be some information about what eigenvalues are used for that non math Ph.D.s can understand. You have the section of "Applications", but it is filled with technical terms like "covariance matrix", which alienates a lot of people. I don't suggest that you delete the current "Applications" section, but rather write something that is easier to read, prefably in the top of the article. The last problem I can think of is that the lead section introduces facts that are not otherwise in the rest of the article. --Maitch 10:57, 18 August 2006 (UTC)

Ok, I didn't feel well and kind of left the scene afer posting this for a few days there. 1) On the cites, the point that straight-up scientific facts do not generally need to be cited is well-taken. I don't know what the Wikipedia policy is on this. However, I do think there need to be some cites, as indicated by Maitch above, for example. And what exactly is wrong with citing textbooks, anyway?? In fact, if I feel up to it, I can help out in that regard. Fact is, though, this article is actually rougher going (and more mechanical) in many places than my linear algebras textbook. I know the nature of the material puts inherent limits on readabiklity and so on. But I think that, even in math, there is a sort of popular science approach to writing versus textbook style. People (even I who know a little bit about this stuff) should not have to click on every single word in the text to find out what the heck is going on. Take this one example of textbook style writing:

Recall that above we defined the geometric multiplicity of an eigenvector to be the dimension of the associated eigenspace, the nullspace of λI − A. The algebraic multiplicity can also be thought of as a dimension: it is the dimension of the associated generalized eigenspace (1st sense), which is the nullspace of the matrix (λI − A)k for any sufficiently large k. That is, it is the space of generalized eigenvectors (1st sense), where a generalized eigenvector is any vector which eventually becomes 0 if λI − A is applied to it enough times successively

It's extremely formal, passive voice "Recall that above the x of the y located" and obviously technical academic writing. I don't mean to pick on anybody here. In fact, I'm not exactly sure that an article this techincal CAN be simplified in a short time. But I think it can be, in principle. Even within the arera of textbooks, there are those which are more and less readable. --Francesco Franco aka Lacatosias 07:18, 19 August 2006 (UTC)

I agree with most of these points. Some parts (e.g., conjugate eigenvalues, eigenfaces) are not standard knowledge and should carry inline citations. I'd even hazard a guess that the last application (to graph theory) is in fact in none of the references. By the way, the problem with inline citations as I see it is that they may distract the reader.

There should be indeed more about the history, which should include the etymology paragraph from the lead. If I have time next week, I might have a look. One of the applications (vibrations) is pretty high up, but should perhaps go even higher, and its practical use more strongly emphasized.

Now, textbook style writing. While everybody agrees that this is inappropriate for an encyclopaedia, it is not clear (at least to me) what "textbook style" actually is. Fortunately, both of you explain what you mean. I'd agree that the fragment shown by Francesco is not brilliant prose. -- Jitse Niesen (talk) 12:34, 19 August 2006 (UTC)

BTW, in terms of content, images and all that important stuff, this is simply an outstanding article. I like the addition of the historical stuff, though. More info like that seems to make it more accessible and less textbookish right off the bat. It makes it less intimidating or something. --Francesco Franco aka Lacatosias 14:32, 22 August 2006 (UTC)

Just in a quick scan, I see missing topics or topics that are referred to only very obliquely, and at least one gross error; overall, the organization needs re-thinking, too. Defective matrices are not mentioned explicitly, although there are some oblique references to generalized eigenvectors. Simultaneous eigenvectors of commuting matrices, a central concept in physical problems because it ties in to the effect of symmetries on the solution to physical problems (and leads directly into connections with representation theory, e.g. Bloch's theorem and other important results), are not mentioned. It doesn't mention the orthogonality of Hermitian-matrix eigenvectors. Left and right eigenvectors are not defined. The generalized eigenvalue problem seems awfully underdescribed to me, and probably needs an article of its own (e.g. to give analogous properties to ordinary Hermitian eigenproblems, etc.). The normal modes of coupled mechanical oscillators (from coupled pendula to continuous problems like vibrating strings) are probably the most familiar example of eigenmodes of Hermitian matrices/operators, much more familiar than quantum mechanics or stress tensors to most people, and need more description. This is mentioned under "examples" but the explanation is totally wrong: the eigenvalue for standing waves is not the amplitude, it is the frequency (or, more precisely, the frequency squared). Damping corresponds to a complex eigenvalue (complex frequency). —Steven G. Johnson 04:44, 23 August 2006 (UTC)

Just a quick comment on your last point, since I see you changed the article in a way which I think is incorrect. I don't think that the eigenvalue is the frequency squared in that context. The section looks at the eigenvalues of the time evolution matrix. So, if the equation of motion is x'' = Ax, then the eigenvalues of exp(At) are considered. Standard is to look at the eigenvalues of A, and these eigenvalues correspond to the frequencies. I think the section needs to be rewritten completely to take a more standard approach. It's an important section, because it's probably the most familiar application of eigenvalues and also historically important. -- Jitse Niesen (talk) 03:01, 24 August 2006 (UTC)

That's a totally nonstandard way to frame the normal-mode eigenvalue problem. And anyway, exp(At) is not the time-evolution operator since this is a second-order problem (the initial value x(0) alone does not determine the later behavior); you need to write it as a first-order problem (of twice the size) to use a matrix exponential. —Steven G. Johnson 15:07, 24 August 2006 (UTC)

See also my discussion on the Talk page. I stand by my comment that the current discussion is erroneous, or at least grossly misleading. —Steven G. Johnson 15:41, 24 August 2006 (UTC)

That was my idea to introduce the swinging rope as an eigenfunction of the time evolution operator. I think it is better that introducting it as an eigenfunction of the Hamiltonian (S=eiHt{\displaystyle S=e^{iHt}}) because the concept of Hamiltonian is more complex to the layman than the time evolution operator. Like this we can use the nice applet which illustrates the point quite well: one should see directly that the rope remains proportional to itself as time passes by. Vb 08:54, 24 August 2006 (UTC)

This has nothing to do with whether you can use the applet. The point that the oscillation is a constant profile multipled by a sinusoidal oscillation is, of course, still true. Just that the eigenvalue is the frequency, not the "amplitude", which makes little sense as I explained. —Steven G. Johnson 15:49, 24 August 2006 (UTC)

Let's take this to the talk page. Regarding the other points you raised: exactly what to mention is a bit a matter of personal preferences. I agree with defectiveness, I don't understand what you're saying about simultaneous eigenvectors (I know they're important in quantum mechanics, and perhaps the connection between measurement and eigenvectors is worth mentioning, but you seem to be talking about something else), and I think the generalized eigenvalue problem is discussed in sufficient detail (though a separate article would of course be nice). Anyway, it's easy to think of topics to add, but we also have to keep the article within a reasonable size. -- Jitse Niesen (talk) 12:13, 25 August 2006 (UTC)

Oh, you don't understand what he's saying about simultaneous eigenvectors, eh?? Seriosly, is this the same thing as quantum superposition?? Good 'eavens!! --Francesco Franco aka Lacatosias 12:19, 25 August 2006 (UTC)

When two Hermitian/unitary (or at least non-defective?) operators or matrices commute, a set of eigenvectors can be chosen that are eigenvectors of both operators simultaneously. This is a fundamental fact about eigenvectors, not limited to simultaneous observables in quantum mechanics, and its consequences are far-reaching. As one example, when an eigenproblem corresponding to some system (whether quantum or classical) has a physical symmetry (e.g. it is rotationally invariant), this means that the unitary transformation operator corresponding to the symmetry commutes with the eigenproblem. This means that the eigenvectors of the system can be chosen to be eigenvectors of the symmetry; more generally, one can show that the eigenvectors of the system can be chosen to transform as irreducible representations of the symmetry group. (The simplest example of this is that the eigenvectors of a system with a mirror symmetry, a cyclic group of order 2, are either even or odd. As another example, matrices invariant under cyclic shifts are diagonalized by the discrete Fourier transform.) This is perhaps the key theorem for the consequences of symmetry on such a problem. —Steven G. Johnson 15:40, 25 August 2006 (UTC)

I agree that there are only so many topics that one can add. However, an article on eigenproblems should at least list (in abbreviated form) all of the relevant definitions and general properties of eigenproblems. Of course, many of these will link to other articles for the detailed explanation, but there should be a summary or at least a mention. (Think of it this way: the article should contain at least as much information as the table of contents of a book on eigenproblems.) Some omissions, like orthogonality of eigenvectors for Hermitian/unitary operators, and the definition of left/right eigenvectors for non-symmetric problems, are particularly glaring to me. As for the generalized eigenproblem, my problem is that there is no explanation whatever as to why it is useful, just a baldfaced claim that it is "preferable". (e.g. if A and B are Hermitian and B is positive-definite, then one can prove analogues of the usual real eigenvalues, orthogonal eigenvectors, etc., properties from the generalized eigenproblem. These properties are not apparent if one writes it in B-1A non-Hermitian form. On the other hand, if A and B are near-singular, the generalized eigenproblem can be inherently numerically unstable, regardless of how it is solved [1].) —Steven G. Johnson 15:57, 25 August 2006 (UTC)

Move to FARC for more work. The article has a lot of prose and syntactical problems, too many stubby sections, poor layout of sentences vs. formulas, and needs a better copyedit for basic items like punctuation. It seems like a native English-speaker should do a thorough copyedit. Some examples (there are more):

The exponential growth or decay provides an example of a continuous spectrum and the vibrating string an example above.

However, in the case we only look for the bound state solutions of the Schrödinger equation, as is usually the case in quantum chemistry, we look for ΨE within the space of square integrable functions.

In this notation. the Schrödinger equation is

H|ΨE⟩=E|ΨE⟩{\displaystyle H|\Psi _{E}\rangle =E|\Psi _{E}\rangle }

and call |ΨE⟩{\displaystyle |\Psi _{E}\rangle } an eigenstate of H (sometimes written H^{\displaystyle {\hat {H}}} in introductory textbooks) which is a self adjoint operator, the infinite dimensional analog of Hermitian matrices (see Observable).

Tighten up weasle words:

In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent of the orbitals and their eigenvalues.

Redundancies:

In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.

Very hard to read, and not because of the math, but because of the English.

Status? This article has received some work since I last looked at it (diff), but the English/prose issues haven't been addressed. I don't have time to work on it: I'm wondering if we can scare up someone to do a serious copyedit of the English/grammar/prose issues? Sandy 13:15, 12 September 2006 (UTC)

Comment. Remove I'm the one who nominated this, so I suppose I should say something. I'd hate to vote this one down because I really do think it contains much high-quality content and is very informative in places. Having said that, it looks like no one is attempting to address any of the objections that have been posted. Nothing is being done at all!! Don't know what to make of the situation.--Francesco Franco aka Lacatosias 08:00, 13 September 2006 (UTC)

Changed to remove. I REALLY hate to do this. But here are my reasons: Since I made the previous comment, the only things that have been addressed are very superfical and mostly done by me (e.g. the examples listed by SandyGeorgia). I just don't see any genuine effort here. I realize that people are very busy, this is all voluntary work, etc., but we have to judge the overall progress of the article.

There is very little change and the article has been out here for quite some time. The discussion also seems to have been sidetracked, to some extent, onto issues of content which I am not qualified to judge.--Francesco Franco aka Lacatosias 07:07, 18 September 2006 (UTC)

I asked someone, who may have time over the weekend, to try to do some work on it. Sandy 03:09, 14 September 2006 (UTC)

Remove, reluctantly, because there's good in this article. This is a difficult topic for most readers, so there's a strong need to make it as simple and clear as possible. It would help if the prose were written to a uniformly "professional" standard, as required, and were more accessible in places. Here are examples of glitches in the prose that make it yet more difficult for our readers:

"The spectral theorem depicts the importance of the eigenvalues and eigenvectors for characterizing a linear transformation in a unique way." No, a theorum doesn't depict; it might describe. Why not "... for uniquely characterizing a linear transformation."?

"A standing wave in a rope fixed at its boundaries can be seen as an example of an eigenvector,..."—Why the hedging? Can't it be "A standing wave in a rope fixed at its boundaries is an example of an eigenvector,..."?

Those examples, and the point about stubby paras, indicate that the entire text needs copy-editing. Tony 11:23, 15 September 2006 (UTC)

Status: Given that this received extensive comments in the first period, I've gone and notified four users that time's up. Hopefully a few more comments will role in. Marskell 19:06, 17 September 2006 (UTC)

Comment: As per Sandy's request, I took a look. The mathematics is mostly sound, but I would be inclined to make substantial modifications to the overall structure, since right now the flow is a bit rough. The intro and history are excellent, but as soon as we hit the definitions, it's a bit too much too fast. It might be better to say something like: "Given a physical or mathematical problem, one may think of eigenvectors as basic waves for this problem. The eigenvalue corresponding to an eigenvector is a measure of the importance of this basic wave. For example..." and then you move everything under "applications", the spectrum of chlorine and maybe the stuff under "examples" in that section. You will have to work on the text to make it go smoothly and shorten it. I would stick all the math in a second section and I might shorten it. The idea of this structure is that in math papers, you put the big ideas in the first few pages of the paper, because nobody reads past page 3 anyway. The thing about math is that you can't write a paragraph that is simultaneously useful to laypeople and experts. You can write A Brief History of Time or you can write Lectures on physics, but there isn't much in between. I have noticed that lots of people use Wikipedia math articles as starting points, if you strip all the math out, they won't be able to do that anymore.

Unfortunately, I don't have time to do this substantial rewrite (papers are due and all.)

I have no opinion of whether this article meets Wikipedia Featured Article standards.

Remove. It doesn't look like anyone is willing to take on this article and do the major rewrite needed: Loisel's and Tony's reviews confirm what I see in the article; that is, the problems are not in the math, but in the prose and the article organization. Improving this article involves more than copyediting the prose issues: it really needs an overall rewrite and restructuring. Sandy 15:54, 19 September 2006 (UTC)